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4. Let V be a vector space over a field F. Let G be a finite group which acts linearly on V (this means that g(xv) = X(gv) and g(v + w) = v + w for all X € F, v, w € V, in addition to the axioms of a group axiom lv = v and (gh)v = g(hv).)

(a) Suppose that |G| #0 in F. Let x E V. Prove that gx = x for all g E G if and only if there exists y EV such that X = 1 |G| Σgy. gEG (b) Let F = Q and V = Q(2, i). Let s denote complex conjugation and r the field automorphism of V with r(2) = 2 and r(i) = i. Use the previous formula with y = = 2 to come up with elements of V fixed by the order two subgroups <rs > and <p³ s>. (c) Find the minimal polynomials over Q of the elements you found in the previous part.